In CLQG (Covariant LQG [1]}), given the Hilbert spaces and
quantum algebras at the boundary, it seems one can only calculate
things like (quantum) properties of the induced spatial \(3-\) d metric
(areas, volumes, curvature) and the extrinsic curvature of the hypersurface,
and not quantum durations (for example, in the analysis done for the
extrinsic coherent states, the area and the extrinsic curvature peak
at those states, but time is introduced as an already peaked
external parameter.) There are some extensions that introduce timelike
boundaries, but, for a time evolution, we need to be able to introduce
durations also in the interior of the process; furthermore,
the commutation relations for variables whose weight functions have
a spacetime support, and in regions in spacetime that are such
that one is in the causal future of the other, are non-trivial (this
is the important result of the so-called covariant Poisson brackets),
and are not accounted for by the part of the commutation relations
that arise purely due to the structure (kinematical in nature)
of the group in the Yang-Mills-like/connection variables used in these
formulations [2]}, since the dependence over the causality
comes from the hyperbolic character of the dynamics and the mentioned
issue with the supports, and then it's indifferent if we take the
Yang-Mills/connection group to be \(SU(2)\)  (spacelike boundary) or
\(SU(1,1)\)  (timelike boundary), that is, this causal part will be
present even if the field variable were a scalar field; as argued
in the paper, and from relational considerations, we consider this
a key issue in the quantization of duration, which makes it quite
different from the quantization of the spatial metric and that it
cannot thus be completely obtained by tweaks on the already existing
methods for the latter. In particular, the extensions of CLQG give
the area of a timelike triangle for vectors (one timelike and the
other spacelike) whose origin is at the same vertex (evidently, the
causal part is irrelevant, since the origin of both vectors is the
same point, and then only the group part enters to play there), while
TQG describes the same area but from the point of view of the causal
connection between the initial and enpoint of the timelike vector
(thus, the causal part is relevant for that); both refer to a same
event and as a finite process, but TQG gives a more dynamical characterization
of it while CLQG givesNote that the dynamical transition in CLQG between the initial and
final points of the previous timelike vector is not the same thing
as the process described by TQG between them: the transition in CLQG
is actually a transition between two processes, the latter ones each
of kinematical area given by the triangles at the points in question,
while the events in TQG describe dynamically (via the causal relation
between the initial and final points) the processes corresponding
to each triangle. The dynamical transitions in CLQG cannot
describe this latter dynamical characterization for each individual
triangle because they are transitions from state to state, triangle
to triangle. The transition described in CLQG corresponds to the transition
between events in TQG, but where each theory describes different aspects
of it (in CLQG, is the dynamics between the kinematics of each event,
while in TQG is the dynamics between the dynamics of each.) We would
say that it's actually TQG the theory that gives actual physical entity
to those events since it describes them as dynamics. its internal kinematical degrees of freedom. Thus, the corresponding
quantum algebra for duration is a composition of the group and causal
aspect of the classical Poisson bracket. From this causal part, unique
to TQG, one gets an universal decay for the transition that
explains the features of duration at the macro scale (this decay is
always there and is independent of the details of the other parts
of the transitions); this part of the transition for duration is ignored
(or considered to have already peaked) in treatments that rely heavily
in coherent states adapted only to the group part.
